Immersion and invariance orbital stabilization of underactuated mechanical systems with collocated pre-feedback

TL;DR

This paper introduces a novel control approach combining partial linearization and immersion-invariance techniques to stabilize oscillations in underactuated mechanical systems with collocated pre-feedback, despite the loss of Euler-Lagrange structure.

Contribution

It presents a constructive, smooth controller design for orbital stabilization of underactuated systems with collocated pre-feedback, addressing structural challenges.

Findings

Successfully stabilizes oscillations in underactuated systems

Provides a constructive, analytic controller formulation

Characterizes system class with verifiable assumptions

Abstract

In this note we study the generation of attractive oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, i.e., a partial linearizing state feedback, and an immersion and invariance orbital stabilization controller. The first step is adopted to simplify analysis and design, however, bringing an additional difficulty that the model loses its Euler-Lagrange structure after the collocated pre-feedback. To address this, we propose a constructive solution to the orbital stabilization problem via a smooth controller in an analytic form, and the model class identified in the paper is characterized via some easily apriori verifiable assumptions on the inertia matrix and the potential energy function.